Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x+8y &= 2 \\ 4x+9y &= 5\end{align*}$
Explanation: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $4$ and the bottom equation by $5$ $\begin{align*}-20x+32y &= 8\\ 20x+45y &= 25\end{align*}$ Add the top and bottom equations. $77y = 33$ Divide both sides by $77$ and reduce as necessary. $y = \dfrac{3}{7}$ Substitute $\dfrac{3}{7}$ for $y$ in the top equation. $-5x+8( \dfrac{3}{7}) = 2$ $-5x+\dfrac{24}{7} = 2$ $-5x = -\dfrac{10}{7}$ $x = \dfrac{2}{7}$ The solution is $\enspace x = \dfrac{2}{7}, \enspace y = \dfrac{3}{7}$.